This page is for developing preliminary notes or making typographical experiments, etc. It may be edited by anybody, anytime. But you don’t necessarily need to delete other people’s ongoing notes here in order to add your own. In any case, overwritten edits may always be recovered from the page history.
In representation theory, the notion of twisted intertwiners is a generalization of that of intertwiners.
References using the exact terminology “twisted intertwiner” include Fuchs & Schweigert 2000a (7.2), 2000b (2.2), Felder, Fröhlich, Fuchs & Schweigert 2000 (4.7), but the notion itself is older and may be known under other names.
For a group and a pair of linear representation on vector spaces , , respectively (all this generalizes straightforwardly to algebras and their modules), a twisted intertwiner
is
an automorphism of the group
such that for all we have
Given a pair of twisted intertwiners their composition is given by composing their components separately.
Writing for the delooping groupoid of and for the category of vector spaces (over a given ground field), the ordinary category Rep of -representations and ordinary intertwiners between these is equivalently the functor category
In contrast, the category of -representations with twisted intertwiners between them is the iso-comma (2,1)-category between and , whose 1-morphisms are diagrams in Grpd of this form:
and whose 2-morphisms are the evident paper-cup pasting diagrams
Giovanni Felder, Jürg Fröhlich, Jürgen Fuchs, Christoph Schweigert: The geometry of WZW branes, J. Geom. Phys. 34 (2000) 162-190 [doi:10.1016/S0393-0440(99)00061-3, arXiv:hep-th/9909030]
Jürgen Fuchs, Christoph Schweigert: Symmetry breaking boundaries II. More structures; examples, Nucl. Phys. B 568 (2000) 543-593 [arXiv:hep-th/9908025, doi:10.1016/S0550-3213(99)00669-0]
Jürgen Fuchs, Christoph Schweigert: Lie algebra automorphisms in conformal field theory, in Conference on Infinite Dimensional Lie Theory and Conformal Field Theory (May 2000) [arXiv:math/0011160]
Particle-hole symmetry.
Original
may be understood as for majority spin polarization with for the minority polarization (Haldane 1983 doi:10.1103/PhysRevLett.51.605)
Here
“” means that the corresponding combination of is not admissible, in that is odd or the Gauss sum .
“” means that
So:
in the column of all odd are excluded and all even are admissible,
in the column of all odd are excluded and exactly every second even is admissible,
in rows of odd the odd are excluded, and for even the result is admissible iff the Jacobi symbol .
Since the Jacobi symbol vanishes iff , this means that, at least for odd , exactly only the reduced fractions appear.
Last revised on April 2, 2025 at 17:14:41. See the history of this page for a list of all contributions to it.